Facets in the Vietoris--Rips complexes of hypercubes

TL;DR

This paper studies the facets of Vietoris--Rips complexes of hypercubes, revealing their complex dimension structure and non-trivial homology, with results linked to Hadamard matrices.

Contribution

It demonstrates that the number of facet dimensions grows super-polynomially with scale and establishes non-trivial homology under certain conditions, connecting combinatorics and topology.

Findings

Number of facet dimensions is super-polynomial in scale r for large n.

$(2r-1)$-th homology is non-trivial when Hadamard matrices of order 2r exist.

Results connect hypercube complexes with Hadamard matrix properties.

Abstract

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.